Maxwell's equations in integral form are very useful for obtaining
a physical understanding of how the electric and magnetic fields
relate to their sources. In fact, the differential forms of
Maxwell's equations are a limiting case of the integral forms,
which makes the integral forms a necessity for gaining physical
intuition and insight into the mathematical relationships that
make up Maxwell's equations. It is difficult to understand the
differential forms with out first understanding the integral
forms. In the development of Maxwell's equations over time, the
equations were first quantified and written out in their integral
forms and then converted to differential form. This said, the
problems solved by the integral forms are somewhat limited and are
usually utilized to solve problems that posses complete symmetries
(such as rectangular, cylindrical, spherical symmetries)
\cite{Balanis:1989}.

By using the \emph{divergence theorem} and \emph{Stokes theorem},
the differential forms can be converted to the integral forms.
Likewise, by using the reverse approach, the integral forms can be
transformed to the differential forms, by integrating over a
volume or surface element and by taking the limiting case as the
the volume and surface element approach zero. The \emph{divergence
theorem} in mathematical form is written as,
\begin{subequations}
\begin{equation}\label{eqn:divergencetheorem}
    \oint_S\vec{A}\cdot{d\vec{s}}=\int_V\nabla\cdot\vec{A}\;dv\qquad\text{Divergence Theorem}
\end{equation}
which simply states that given a volume $V$ enclosed by a surface
$S$, the total out-flowing flux of $\vec{A}$ through the surface
$S$, is equal to the divergence of the vector $\vec{A}$ integrated over
the volume $V$. \emph{Stoke's theorem} is expressed as,
\begin{equation}\label{eqn:stokestheorem}
    \oint_C\vec{A}\cdot{d\vec{\ell}}=\int_S\nabla\times\vec{A}\cdot\;d\vec{s}\qquad\text{Stoke's Theorem}
\end{equation}
\end{subequations}
which states that given a surface $S$ enclosed by the curve $C$,
the line integral of the vector $\vec{A}$ around the closed curve
$C$ is equal to the curl of the vector $\vec{A}$ integrated over
the surface $S$.

Applying the \emph{divergence theorem}
(\ref{eqn:divergencetheorem}) to the differential form of Gauss's
electric field law (\ref{eqn:gefl}) and to Gauss's magnetic field
law (\ref{eqn:gmfl}) yields the integral form equivalents,
\begin{subequations}
\begin{align}
    \oint_S\vec{\mathcal{D}}\cdot{d\vec{s}}&=\int_V\rho_{ev}\;dv=Q_e&&\text{Gauss's E-Field Law}\label{eqn:geflI}\\
    \oint_S\vec{\mathcal{B}}\cdot{d\vec{s}}&=\int_V\rho_{mv}\;dv=0&&\text{Gauss's H-Field Law}\label{eqn:ghflI}
\end{align}
\end{subequations}
In words, (\ref{eqn:geflI}) says that the total electric charge
found in the volume $V$ enclosed by the surface $S$ is equal to
the total out-flowing flux through the surface $S$. Equation
(\ref{eqn:ghflI}) has a similar interpretation but the fictitious
magnetic charge density is equal to zero.  This means that all of
the magnetic flux  going out of the surface $S$ returns to
form closed loop flux lines. In order for this to be true, the
magnetic charges must occur in pairs and cannot be separated.

Applying Stokes theorem (\ref{eqn:stokestheorem}) to the
differential from of Faraday's law (\ref{eqn:fl}) and Ampere's law
(\ref{eqn:al}) yields the integral form equivalents.
\begin{align}
    \oint_C\vec{\mathcal{E}}\cdot{d\vec\ell}&=-\frac{\partial}{\partial{t}}\int_S\vec{\mathcal{B}}\cdot{d\vec{s}}\;-\!\int_S\vec{\mathcal{M}}\cdot{d\vec{s}}&&\text{Faraday's Law}\label{eqn:flI}\\
    \oint_C\vec{\mathcal{H}}\cdot{d\vec\ell}&=\frac{\partial}{\partial{t}}\int_S\vec{\mathcal{D}}\cdot{d\vec{s}}\;+\!\int_S\vec{\mathcal{J}}\cdot{d\vec{s}}&&\text{Ampere's Law}\label{eqn:alI}
\end{align}
Equation (\ref{eqn:flI}) states that the sum of the negative time changing magnetic flux and fictitious magnetic current
flux through a given area $S$, who's perimeter is the closed loop
$C$, is equal to the vector line integral of the magnetic field
intensity. Equation (\ref{eqn:alI}) has a similar interpretation.

The negative sign in (\ref{eqn:flI}) is a physical manifestation
that the induced curl of the electric field intensity is in a
direction so as to oppose the change in the magnetic flux
producing it. This physical observation is known as \emph{Lenz's
law} \cite{Iskander:1992}.

The differential form of the continuity equation
(\ref{eqn:continuityeqn}) can be changed into integral form by
using the divergence theorem (\ref{eqn:divergencetheorem}). This
is written as,
\begin{equation}\label{eqn:continuityeqnI}
    \oint_S \vec{\mathcal{J}}\cdot
    d\vec{s}=-\frac{\partial}{\partial t}\int_V q_{ev}\;dv\qquad\text{Continuity Equation}
\end{equation}
In order to obtain a physical understanding of
(\ref{eqn:continuityeqnI}), consider a volume $V$ enclosed by a
surface $S$ with a certain amount of charge inside the surface.
Charge flowing out of the surface (the current flux) will cause a
reduction of charge (or charge density) in the volume. In other
words, the total current flux flowing out of the enclosed surface
is equal to the negative rate of change of current in the volume.

It should be noted that this is the equation that Maxwell used to
introduce his displacement current term in Ampere's law. Although,
this equation can be derived from Maxwell's equations,
historically it happened in the other order.
